Abstract

The main purpose of this paper is to introduce a method of successive approximations in terms of a hybrid of Taylor series and a block-pulse function, which is given for solving nonlinear fuzzy Fredholm integral equations of the second kind and the error estimate of the approximation solution. Finally, two numerical examples are presented to show the accuracy of the proposed method.

Keywords

1 Introduction

The study of fuzzy integral equations (FIE) from both theoretical and numerical points of view has been developed in recent years after a distinct study of the existence of a unique solution for fuzzy Fredholm integral equations had been carried out in [1]. The Banach fixed point theorem is the main tool in studying the existence and uniqueness of the solution for fuzzy integral equations which can be found in [2–4]. Numerical procedures for solving fuzzy integral equations of the second kind, based on the method of successive approximations and other iterative techniques, have been investigated in [5–9]. The Adomian decomposition method is used in [10–12], as well as quadrature rules, fuzzy Bernstein polynomials, fuzzy wavelet, and Nystorm techniques were applied to fuzzy integral equations of the second kind in [13–18].

Recently, Bica and Popescu [19, 20] applied the method of successive approximations for the fuzzy Hammerstein integral equation. Ezzati and Ziari [21] proved the convergence of the method of successive approximations for solving nonlinear fuzzy Fredholm integral equations of the second kind, and they proposed an iterative procedure based on the trapezoidal quadrature. Mirzaee [22] obtained an approximate solution for the linear Fredholm fuzzy integral equations of the second kind by the hybrid of block-pulse function and Taylor series (HBT).

In this paper, we approximate the fuzzy function by the hybrid Taylor and block-pulse functions (HBT) and estimate the error approximation. Also, an iterative procedure is constructed based on HBT for solving nonlinear Fredholm fuzzy integral equations,

\([u]_{0}=\overline{\{x\in{\mathbb{R}}: u(x)>0\}}\) is a compact interval, where \(\overline{A}\) is the closure of the set A.

The set of all fuzzy real numbers is denoted by \({{\mathbb{R}}}_{{\mathcal{F}}}\). Any real number \(a\in{\mathbb{R}}\) can be interpreted as a fuzzy number \(\tilde{a}={\chi}_{\{a\}}\) and therefore \({\mathbb{R}}\subset{{\mathbb{R}}}_{{\mathcal{F}}}\).

For \(0< r\leq1\), we denote \({[u]}_{r}=\{x\in{\mathbb{R}}:u(x)\geq r\}\) the r-level (or simply the r-cut) set of u which is a closed interval (see [24]) and \({[u]}_{r}=[{\underline{u}}_{r},{\overline {u}}_{r}]\), \(\forall r\in[0,1]\). This leads to the usual parametric representation of a fuzzy number.

The standard Hukuhara difference (H-difference \({\ominus}_{H}\)) is defined by \(u\ominus_{H}v=w\Longleftrightarrow u=v\oplus w\); if \(u\ominus_{H}v\) exists, its r-cuts are \({ [u\ominus _{H}v ]}_{r}= [{\underline{u}}_{r}-{\underline{v}}_{r},{\overline {u}}_{r}-{\overline{v}}_{r} ]\). It is well known that \(u\ominus _{H}u=\tilde{0}\) for all fuzzy numbers u, but \(u\ominus u\ne \tilde{0}\).

Lemma 2.3

Remark 2.1

The properties (iv) in Lemma 2.2 suggest the definition of a function \(\Vert \cdot \Vert :{{\mathbb{R}}}_{{\mathcal{F}}}\to {\mathbb{R}}\) by \(\Vert u\Vert =D (u,\tilde{0} )\), which has the properties of the usual norms. In [5] the properties of this function are presented as follows:

It is easy to show that (i) and (ii) are both valid if and only if w is a crisp number.

In terms of r-cuts we have \({ [u\ominus_{gH}v ]}_{r}= [\min\{{\underline{u}}_{r}-{\underline{v}}_{r},{\overline {u}}_{r}-{\overline{v}}_{r}\},\max\{{\underline{u}}_{r}-{\underline {v}}_{r},{\overline{u}}_{r}-{\overline{v}}_{r}\} ]\), and if the H-difference exists, then \(u\ominus_{H}v=u\ominus_{gH}v\); the conditions for the \(w=u\ominus_{gH}v\in\mathbb{R}_{{\mathcal{F}}}\) are

Proposition 2.2

Let\(u ,v\in{{\mathbb{R}}}_{{\mathcal{F}}}\)be two fuzzy numbers; then

(a)

if the gH-difference exists, it is unique,

(b)

\(u\ominus_{gH}v=u\ominus_{H}v\)or\(u\ominus _{gH}v=- (v\ominus_{H}u )\)whenever the expressions on the right exist; in particular, \(u\ominus_{gH}u=u\ominus_{H}u=\tilde{0}\),

(c)

if\(u\ominus_{gH}v\)exists in the sense of (i), then\(v\ominus_{gH}u\)exists in the sense of (ii) and vice versa,

(d)

\((u\oplus v )\ominus_{gH}v=u\),

(e)

\(\tilde{0}\ominus_{gH} (u\ominus_{gH}v )=v\ominus_{gH}u\),

(f)

\(u\ominus_{gH}v=v\ominus_{gH}u=w\)if and only if\(w=-w\); furthermore, \(w=\tilde{0}\)if and only if\(u=v\).

Definition 2.4

For any fuzzy-number-valued function \(f:I\subset {\mathbb{R}}\longrightarrow{{\mathbb{R}}}_{{\mathcal{F}}}\) we can define the functions \({\underline{f}}_{r},{\overline{f}}_{r}:I\subset{\mathbb{R}}\longrightarrow{\mathbb{R}}\), \(r\in[0,1]\) by \({\underline{f}}_{r} (t )={\underline{ (f(t) )}}_{r}\), \({\overline{f}}_{r} (t )={\overline{ (f(t) )}}_{r}\), \(\forall t\in[0,1]\). These functions are the left and right r-level functions of f.

Definition 2.5

A fuzzy-number-valued function \(f:[a,b]\longrightarrow{{\mathbb{R}}}_{{\mathcal{F}}}\) is said to be continuous at \(t_{0}\in[a,b]\) if for each \(\varepsilon>0\) there is \(\delta>0\) such that \(D (f (t ),f (t_{0} ) )<\varepsilon\) whenever \(\vert t-t_{0}\vert <\delta\). If f is continuous for each \(t\in[a,b]\) then we say that f is fuzzy continuous on \([a,b]\). A fuzzy number \(u\in{{\mathbb{R}}}_{{\mathcal{F}}}\) is upper bound for a fuzzy-number-valued function \(f:[a,b]\longrightarrow{{\mathbb{R}}}_{{\mathcal{F}}}\) if \({\underline { (f(t) )}}_{r}\le{\underline{u}}_{r}\) and \({\overline{ (f(t) )}}_{r}\le{\overline{u}}_{r}\) for all \(t\in[a,b]\). A fuzzy number \(u\in{{\mathbb{R}}}_{{\mathcal{F}}}\) is a lower bound for a fuzzy-number-valued function \(f:[a,b]\longrightarrow{{\mathbb{R}}}_{{\mathcal{F}}}\) if \({\underline{u}}_{r}\le{\underline{ (f(t) )}}_{r}\) and \({\overline{u}}_{r}\le{\overline{ (f(t) )}}_{r}\) for all \(t\in[a,b]\). A fuzzy-number-valued function \(f:[a,b]\longrightarrow{{\mathbb{R}}}_{{\mathcal{F}}}\) is said to be bounded if it has a lower and an upper bound.

Remark 2.2

The above definition of the boundedness of a fuzzy-number-valued function can be expressed in the following equivalent form: \(f:[a,b]\longrightarrow{{\mathbb{R}}}_{{\mathcal{F}}}\) is bounded iff there is \(M\ge0\) such that \(D (f (t ),\tilde{0} )\le M\) for all \(t\in[a,b]\). The constant M can be chosen as \(M\ge{\max \{\vert {\underline{u}}_{0}\vert ,\vert {\overline{u}}_{0}\vert \} }\).

Lemma 2.4

If\(f:[a,b]\longrightarrow{{\mathbb{R}}}_{{\mathcal{F}}}\)is continuous then it is bounded and its supremum\(\sup_{t\in[a,b]} f(t)\)must exist and is determined by\(u\in{{\mathbb{R}}}_{{\mathcal{F}}}\)with\({\underline{u}}_{r}= \sup_{t\in[a,b]} {\underline{f}}_{r} (t )\)and\({\overline {u}}_{r}=\sup_{t\in[a,b]} {\overline{f}}_{r} (t ) \). A similar conclusion for the infimum is also true.

Let \(C_{{\mathcal{F}}}[a,b]\), be the space of fuzzy continuous functions with the metric

which is called the uniform distance between fuzzy-number-valued functions. We see that \((C_{{\mathcal{F}}}[a,b],D^{*} )\) is a complete metric space and using Lemma 2.2 and Lemma 2.3 we can derive the corresponding properties of the metric \(D^{*}\).

Definition 2.6

A fuzzy-number-valued function \(f:[a,b]\longrightarrow{{\mathbb{R}}}_{{\mathcal{F}}}\) is said to be uniformly continuous on \([a,b]\), if for each \(\varepsilon>0\) there is \(\delta>0\) such that \(D(f(t),f(t'))<\varepsilon\) whenever \(t,t'\in [a,b]\) with \(|t-t'|<\delta\).

Definition 2.7

A fuzzy-number-valued function \(f:[a,b]\longrightarrow{{\mathbb{R}}}_{{\mathcal{F}}}\) is said to level-continuous at \(t_{0}\in[a,b]\), if \(\lim_{t\to t_{0}} {\underline{ (f(t) )}}_{r} ={\underline{ (f(t_{0}) )}}_{r}\) and \(\lim_{t\to t_{0}} {\overline{ (f(t) )}}_{r} ={\overline{ (f(t_{0}) )}}_{r}\) for all \(r\in[0,1]\). If f is level-continuous at each \(t\in[a,b]\), then we say that f is level-continuous on \([a,b]\).

It is obvious that the continuity of a fuzzy-number-valued function implies the level-continuity, but the converse does not hold. However, the boundedness property holds for both types of continuity.

Definition 2.10

Let \(x_{0}\in\,]a,b[\) and h be such that \(x_{0}+h\in\,]a,b[\), then the level-wise gH-derivative (LgH-derivative for short) of a function \(f:\,]a,b[\,\to{{\mathbb{R}}}_{{\mathcal{F}}}\) at \(x_{0}\) is defined as the set of interval-valued gH-derivatives, if they exist,

If \(f'_{LgH}{ (x_{0} )}_{r}\) is a compact interval for all \(r\in [0,1]\), we say that f is level-wise generalized Hukuhara differentiable (LgH-differentiable for short) at \(x_{0}\) and the family of intervals \(\{f'_{LgH}{ (x_{0} )}_{r}:r\in[0,1] \}\) is the LgH-derivative of f at \(x_{0}\), denoted by \(f'_{LgH} (x_{0} )\).

Consequently, LgH-differentiability, as is level-wise continuity, is a necessary condition for gH-differentiability; but from Eq. (2.1), it is not sufficient.

The next result gives the analogous expression of the fuzzy gH-derivative in terms of the derivatives of the endpoints of the level sets. This result extends the result given in [31, Theorem 5] and it is a characterization of the gH-differentiability for an important class of fuzzy functions.

Proposition 2.4

Let\(f:\,]a,b[\,\to{{\mathbb{R}}}_{{\mathcal{F}}}\)be such that\({ [f(x) ]}_{r}= [{\underline{f}}_{r}(x),{\overline{f}}_{r}(x) ]\). Suppose that the functions\({\underline{f}}_{r}(x)\)and\({\overline{f}}_{r}(x)\)are real-valued functions, differentiable w.r.t. x, uniformly w.r.t. \(r\in [0,1]\). Then the function\(f(x)\)is gH-differentiable at a fixed\(x\in \,]a,b[\)if and only if one of the following two cases holds:

Denote the set of all functions \(f\in C^{n}_{{\mathcal{F}}} [a,b ]\), \(n\ge1\), such that \(f^{ (k )} [a,b ]\to{{\mathbb{R}}}_{{\mathcal{F}}}\) (\(k=0,1,\ldots,n-1 \)) is (i)-gH-differentiable on the interval \([a,b]\) by \(C^{n,1}_{{\mathcal{F}}}[a,b]\).

By Theorem 5.2 of [21], for \(f\in C^{n,1}_{{\mathcal{F}}}[a,b]\) we obtain

Proposition 2.5

If\(f: [a,b ]\to{{\mathbb{R}}}_{{\mathcal{F}}}\)is gH-differentiable (or right or left gH-differentiable) at\(x_{0}\in[a,b]\)then it is level-wise continuous (or right or left level-wise continuous) at\(x_{0}\).

Definition 2.12

Let \(f: [a,b ]\to{{\mathbb{R}}}_{{\mathcal{F}}}\). For \({\triangle}_{n}:a=x_{0}< x_{1}<\cdots <x_{n-1}<x_{n}=b\) a partition of the interval \([a,b]\), we consider the points \({\xi}_{i}\in [x_{i-1},x_{i} ]\), \(i=1,\ldots,n\), and the function \(\delta: [a,b ]\to{{\mathbb{R}}}_{+}\). The partition \(P= \{ ( [x_{i-1},x_{i} ];{\xi}_{i} );i=1,\ldots,n \}\) denoted by \(P= ({\triangle}_{n},\xi )\) is called δ-fine iff \([x_{i-1},x_{i} ]\subseteq ({\xi}_{i}-\delta ({\xi}_{i} ),{\xi}_{i}+\delta({\xi}_{i}) )\). For \(I\in{{\mathbb{R}}}_{{\mathcal{F}}}\), the function f is fuzzy Henstock integrable on \([a,b]\) if for any \(\varepsilon>0\) there is a function \(\delta: [a,b ]\to{{\mathbb{R}}}_{+}\) such that for any partition δ-fine P, \(D (\sum^{n}_{i=1}{ (x_{i}-x_{i-1} )\odot f ({\xi}_{i} ),I} )<\varepsilon \). The fuzzy number I is named the fuzzy Henstock integral of f and will be denoted by \((FH)\int^{b}_{a}{f (t )\,dt}\).

When the function \(\delta:[a,b]\to{{\mathbb{R}}}_{+}\) is constant, then we obtain the Riemann integrability for fuzzy-number-valued functions (see [32]). In this case, \(I\in{{\mathbb{R}}}_{{\mathcal{F}}}\) is called the fuzzy Riemann integral of f on the interval \([a,b]\), denoted by \((FR)\int^{b}_{a}{f(t)\,dt}\). Consequently, the fuzzy Riemann integrability is a particular case of the fuzzy Henstock integrability, and therefore the properties of the integral (FH) will be valid for the integral (FR), too.

Lemma 2.5

Let\(f: [a,b ]\to{{\mathbb{R}}}_{{\mathcal{F}}}\). Then f is (FH) integrable if and only if\({\underline{f}}_{r}\)and\({\overline{f}}_{r}\)are Henstock integrable for any\(r\in[0,1]\). Furthermore, for any\(r\in[0,1]\),

Remark 2.3

If \(f: [a,b ]\to{{\mathbb{R}}}_{{\mathcal{F}}}\) is fuzzy continuous, then \({\underline{f}}_{r}\) and \({\overline{f}}_{r}\) are continuous for any \(r\in[0,1]\) and consequently, they are Henstock integrable. According to Lemma 2.5 we infer that f is (FH) integrable.

Remark 2.4

If \(f: [a,b ]\to{{\mathbb{R}}}_{{\mathcal{F}}}\) is fuzzy continuous, for a partition \(\triangle:a=x_{0}< x_{1}<\cdots <x_{n-1}<x_{n}=b\), according to [24], the fuzzy-Riemann integral has the property

The block-pulse functions on \([0,1)\) are disjoint, so for \(i,j=1,2,\ldots,N\), we have \({\varphi}_{i} (t ){\varphi }_{j} (t )={\delta}_{i,j}{\varphi}_{i} (t )\), where \({\delta}_{i,j}\) is the Kronecker delta, also these functions have the property of orthogonality on \([0,1)\). For more details see [36].

Remark 3.1

4 Fuzzy integral equations

We consider the nonlinear fuzzy Fredholm integral Eq. (1.1), where \(k (s,t )\) is a positive crisp kernel function over the square \(a\le s,t\le b\), \(F (t )\) is a fuzzy-number-valued function and \(G:R_{F}\to R_{F}\) is continuous. We assume that k is continuous and therefore it is uniformly continuous with respect to t and there exists \(M>0\), such that \(M=\max_{a\le s,t\le b} \vert k (s,t )\vert \).

Remark 5.1

6 Numerical examples

In this section, we apply the presented method in Section 4 for solving the fuzzy integral Eq. (4.3) in two examples. The approximate solution is calculated for different values of N, l and m. Also, we compare the numerical solution obtained by using the proposed method with the exact solution. The computations associated with the examples were performed using Mathematica 7.

7 Conclusion

In this paper, we have suggested an iterative procedure by utilizing fuzzy HBT to solve the nonlinear Ferdholm fuzzy integral Eq. (4.3). The error estimate of the approximated function was obtained by using the fuzzy Taylor theorem [34] for the function which is (i)-gH-differentiable. The error estimate of the present method is proved; for getting the best approximating solution of the equation, the number N and the degree of the fuzzy hybrid polynomial l must be chosen sufficiently large. The analyzed example illustrates the ability and reliability of the fuzzy HBT method for Eq. (4.3).

Declarations

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Authors’ Affiliations

(1)

Department of Mathematics, Science and Research Branch, Islamic Azad University